Three vertices of a Parallelogram ABCD are A (1, 2, 3), B (– 1, – 2, – 1) and C (2, 3, 2). Find the fourth vertex D.
Given three consecutive vertices of a parallelogram $A B C D$ are $A(1,2,3), B(-1$, $-2,-1)$ and $C(2,3,2)$
Let the fourth vertex be $D(x, y, z)$.
Now by using midpoint formula,
Midpoint of diagonal $A C=P\left(\frac{1+2}{2}, \frac{2+3}{2}, \frac{3+2}{2}\right)=P\left(\frac{3}{2}, \frac{5}{2}, \frac{5}{2}\right)$
Midpoint of diagonal BD $=\mathrm{P}\left(\frac{\mathrm{x}+(-1)}{2}, \frac{\mathrm{y}+(-2)}{2}, \frac{\mathrm{z}+(-1)}{2}\right)=\mathrm{P}\left(\frac{3}{2}, \frac{5}{2}, \frac{5}{2}\right)$
On comparing we get
$\Rightarrow \frac{x-1}{2}=\frac{3}{2}$
$\Rightarrow x=4$
$\Rightarrow \frac{y-2}{2}=\frac{5}{2}$
$\Rightarrow y=7$
$\Rightarrow \frac{z-1}{2}=\frac{5}{2}$
$\Rightarrow z=6$
$\therefore \mathrm{D}(4,7,6)$ is the fourth vertex.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,